Mean-Field games with absorption and singular controls

The first part of the work is devoted to mean-field games with absorption, a class of games that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit a given boundary. In most of the literature on mean-field games, all players stay in the game until the end of the period, while in many applications, especially in economics and finance, it is natural to have a mechanism deciding when a player has to leave. Such a mechanism can be modelled by introducing an absorbing boundary for the state space. The second part of the thesis, deals with mean-field games of finite-fuel capacity expansion with singular controls. While singular control problems with finite (and infinite) fuel find numerous applications in the economic literature and originated from the engineering literature in the late 60\u2019s, many-player game versions of these problems have only very recently been introduced. They are a natural extension of the single agent set-up and allow to model numerous applied situations. In our work in particular, we make assumptions on the structure of the interaction across players that are suitable to model the so-called goodwill problem. Altogether, the original contribution to the mean-field games literature of the present work is threefold. First, it contributes to the development of mean-field games with absorption, continuing the work of Campi and Fischer (2018) and considerably generalizing the original model by relaxing the assumptions and setting it into a more abstract, infinite-dimensional, framework. Second, it introduces a new set of tools to deal with mean-field games with singular controls, extending the well-known connection between singular stochastic control and optimal stopping to mean-field games. Finally, it also contributes to the numerical literature on mean-field games, by proposing a numerical scheme to approximate the solutions of mean-field games with singular controls with a constructive approach. Overall, this thesis focuses on newly introduced branches of the theory of meanfield games that display a high potential for economic and financial applications, contributing to the literature not only by further developing the existing theory but also by working in directions that make the these models more suitable to application

Multipartite entanglement detection for hypergraph states

We study the entanglement properties of quantum hypergraph states of $n$ qubits, focusing on multipartite entanglement. We compute multipartite entanglement for hypergraph states with a single hyperedge of maximum cardinality, for hypergraph states endowed with all possible hyperedges of cardinality equal to $n-1$ and for those hypergraph states with all possible hyperedges of cardinality greater than or equal to $n-1$. We then find a lower bound to the multipartite entanglement of a generic quantum hypergraph state. We finally apply the multipartite entanglement results to the construction of entanglement witness operators, able to detect genuine multipartite entanglement in the neighbourhood of a given hypergraph state. We first build entanglement witnesses of the projective type, then propose a class of witnesses based on the stabilizer formalism, hence called stabilizer witnesses, able to reduce the experimental effort from an exponential to a linear growth in the number of local measurement settings with the number of qubits

N-player games and mean-field games with smooth dependence on past absorptions.

Mean-field games with absorption is a class of games that has been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly innite-dimensional, dependence on the (non-normalized) empirical sub- probability measure of the survivors' states. This includes the particularly relevant case where the mean-eld interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players' private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry-Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N tends to infinity.Mean-field games with absorption is a class of games that has been introduced in (Ann. Appl. Probab. 28 (2018) 2188–2242) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly infinite-dimensional, dependence on the (non-normalized) empirical sub-probability measure of the survivors’ states. This includes the particularly relevant case where the mean-field interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players’ private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry–Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N → ∞

NN-player games and mean field games of moderate interactions

We study the asymptotic organization among many optimizing individuals interacting in a suitable "moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument

N-player games and mean field games of moderate interactions

We study the asymptotic organization among many optimizing individuals interacting in a suitable “moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument

Mean-Field Games of Finite-Fuel Capacity Expansion with Singular Controls

We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework

Kelly betting with quantum payoff: A continuous variable approach

The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all their initial resources into coherent state amplitude

Kelly Betting with Quantum Payoff: a continuous variable approach

The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all her/his initial resources into coherent state amplitude.Comment: 14 pages, 8 figure

The “Diabetes Comorbidome”: A Different Way for Health Professionals to Approach the Comorbidity Burden of Diabetes

(1) Background: The disease burden related to diabetes is increasing greatly, particularly in older subjects. A more comprehensive approach towards the assessment and management of diabetes’ comorbidities is necessary. The aim of this study was to implement our previous data identifying and representing the prevalence of the comorbidities, their association with mortality, and the strength of their relationship in hospitalized elderly patients with diabetes, developing, at the same time, a new graphic representation model of the comorbidome called “Diabetes Comorbidome”. (2) Methods: Data were collected from the RePoSi register. Comorbidities, socio-demographic data, severity and comorbidity indexes (Cumulative Illness rating Scale CIRS-SI and CIRS-CI), and functional status (Barthel Index), were recorded. Mortality rates were assessed in hospital and 3 and 12 months after discharge. (3) Results: Of the 4714 hospitalized elderly patients, 1378 had diabetes. The comorbidities distribution showed that arterial hypertension (57.1%), ischemic heart disease (31.4%), chronic renal failure (28.8%), atrial fibrillation (25.6%), and COPD (22.7%), were the more frequent in subjects with diabetes. The graphic comorbidome showed that the strongest predictors of death at in hospital and at the 3-month follow-up were dementia and cancer. At the 1-year follow-up, cancer was the first comorbidity independently associated with mortality. (4) Conclusions: The “Diabetes Comorbidome” represents the perfect instrument for determining the prevalence of comorbidities and the strength of their relationship with risk of death, as well as the need for an effective treatment for improving clinical outcomes

Antidiabetic Drug Prescription Pattern in Hospitalized Older Patients with Diabetes

Objective: To describe the prescription pattern of antidiabetic and cardiovascular drugs in a cohort of hospitalized older patients with diabetes. Methods: Patients with diabetes aged 65 years or older hospitalized in internal medicine and/or geriatric wards throughout Italy and enrolled in the REPOSI (REgistro POliterapuie SIMI—Società Italiana di Medicina Interna) registry from 2010 to 2019 and discharged alive were included. Results: Among 1703 patients with diabetes, 1433 (84.2%) were on treatment with at least one antidiabetic drug at hospital admission, mainly prescribed as monotherapy with insulin (28.3%) or metformin (19.2%). The proportion of treated patients decreased at discharge (N = 1309, 76.9%), with a significant reduction over time. Among those prescribed, the proportion of those with insulin alone increased over time (p = 0.0066), while the proportion of those prescribed sulfonylureas decreased (p < 0.0001). Among patients receiving antidiabetic therapy at discharge, 1063 (81.2%) were also prescribed cardiovascular drugs, mainly with an antihypertensive drug alone or in combination (N = 777, 73.1%). Conclusion: The management of older patients with diabetes in a hospital setting is often sub-optimal, as shown by the increasing trend in insulin at discharge, even if an overall improvement has been highlighted by the prevalent decrease in sulfonylureas prescription